1. EQ: How do we describe and summarize random events?
Random Variables
EQ: How do we describe and summarize random events?

2. Random Variable
Definition: A variable whose value is based on the outcome of a random event.
Discrete: we can list all of the outcomes
Ex: Flipping a coin, making a basketball shot
Continuous: all of the outcomes can take a value within a certain interval.
Ex: Height of men, arrival times, any normally distributed variable

3. Discrete RV Grade Distribution
Students in Stat 101 had the following grade distribution: 21% A’s, 43% B’s, 30% C’s, 5% D’s, and 1% F’s.
Define the random variable:
X = the grade of a randomly selected student on a 4 point scale
X = Grade 4 3 2 1
P(X)

4. Using the distribution
X = Grade 4 3 2 1
P(X)
What is the probability of a student receiving an F?
P(X=0) = .01
What is the probability of a student receiving an F or D?
P(X=0 or X=1) = = .06
What is the probability of a student receiving a grade higher than a C?
P(X > 2 ) = = .64
P (X > 3) = = .64

5. Flipping a Coin Random Variable: Probability distribution:
Y = the number of heads after tossing a coin 4 times.
Probability distribution:
Sample Space: {0,1,2,3,4} Y 1 2 3 4
P(Y)

6. Finding P(Y) Identify the number of ways each event can happen:
Total number of ways: 2(2)(2)(2) = 16
Split the 16 ways up among the events
Y = 0:
Y = 1:
Y = 2:
Y = 3:
Y = 4:
TTTT
HTTT, THTT, TTHT, TTTH
HHTT, TTHH,THTH,HTHT,HTTH,THHT
HHHT, HHTH,HTHH,THHH
HHHH

7. Probability of each event
TTTT
THTH
THHH
HHTT
(.5)(.5)(.5)(.5) =
Every outcome has the same probability!
(.5)(.5)(.5)(.5) =
(.5)(.5)(.5)(.5) =
(.5)(.5)(.5)(.5) =

8. Distribution of Y The sum of P(Y) = 1, if not then you did it wrong! Y1 2 3 4
P(Y)
1(.0625)
4(.0625)
6(.0625)
4(.0625)
1(.0625) Y 1 2 3 4
P(Y)
.0625
.25
.375
.25
.0625
The sum of P(Y) = 1, if not then you did it wrong!

9. Answer Questions Probability of tossing at least two heads:1 2 3 4
P(Y)
.0625
.25
.375
.25
.0625
Probability of tossing at least two heads:
P (Y > 2 ) = = .6875
Probability of at least one head:
P (Y > 1) = 1 – P (Y < 1) = 1 – P (Y = 0) = = .9375

10. Example:
Suppose each of four randomly selected customers purchasing a monkey shaving kit at a certain store chooses either an electric (E) or a gas (G) model. Assume that these customers make their choices independently of one another and that, according to Consumer Reports, 40% of all customers select an electric model while 60% tend to choose the gas model.
Find the probability that at least two of the four customers will choose the electric model. Be sure to define your random variable, provide a probability distribution.

11. Define the Random Variable X = the number of people out of 4 who choose the electric model Identify the sample space {0,1,2,3,4} Find the outcomes for each event Use E for electric and G for gas Find the probabilities for each outcome and then the total probability of the event

12. Making the distribution
0 = GGGG 1 = EGGG, GEGG, GGEG, GGGE 2 = EEGG, GGEE, GEGE, EGEG, EGGE, GEEG 3 = EEEG, EEGE, EGEE, GEEE 4 = EEEE
P(0) = GGGG = (.6)(.6)(.6)(.6) = .1296
P(1) = (.4)(.6)(.6)(.6) + (.6)(.4)(.6)(.6) + (.6)(.6)(.4)(.6)+ (.6)(.6)(.6)(.4) = .3456
P(2) = 6(.4)(.4)(.6)(.6) = .3456
P(3) = 4(.4)(.4)(.4)(.6) = .1536
P(4) = (.4)(.4)(.4)(.4) = .0256

13. Distribution X 1 2 3 4
P(X)
.1296
.3456
.1536
.0256
Find the probability that at least two of the four customers will choose the electric model. Be sure to define your random variable, provide a probability distribution.
P(X > 2) = = .5248